State space optimization of finite state machines from the

Yue et al. / Front Inform Technol Electron Eng 2021 22(12):1598-1609 1598

State space optimization of finite state machines from the

viewpoint of control theory*

Jumei YUE†1, Yongyi YAN†‡2, Zengqiang CHEN3, He DENG2 1College of Agricultural Equipment Engineering, Henan University of Science and Technology, Luoyang 471003, China

2College of Information Engineering, Henan University of Science and Technology, Luoyang 471003, China 3College of Artificial Intelligence, Nankai University, Tianjin 300071, China

†E-mail: [email protected]; [email protected]

Received Nov. 6, 2020; Revision accepted Jan. 3, 2021; Crosschecked July 19, 2021; Published online Oct. 23, 2021

Abstract: Motivated by the inconvenience or even inability to explain the mathematics of the state space optimization of finite state machines (FSMs) in most existing results, we consider the problem by viewing FSMs as logical dynamic systems. Borrowing ideas from the concept of equilibrium points of dynamic systems in control theory, the concepts of t-equivalent states and t-source equivalent states are introduced. Based on the state transition dynamic equations of FSMs proposed in recent years, several mathematical formulations of t-equivalent states and t-source equivalent states are proposed. These can be analogized to the necessary and sufficient conditions of equilibrium points of dynamic systems in control theory and thus give a mathematical explanation of the optimization problem. Using these mathematical formulations, two methods are designed to find all the t-equivalent states and t-source equivalent states of FSMs. Further, two ways of reducing the state space of FSMs are found. These can be implemented without computers but with only pen and paper in a mathematical manner. In addition, an open question is raised which can further improve these methods into unattended ones. Finally, the correctness and effectiveness of the proposed methods are verified by a practical language model. Key words: Finite state machines; Finite-valued systems; Logical systems; Logical networks; Semi-tensor product of matrices;

Space optimization https://doi.org/10.1631/FITEE.2000608 CLC number: TP13

1 Introduction

A finite state machine (FSM) is a mathematical model of computation and is widely used in some areas such as artificial intelligence (Voeten and van Zaanen, 2018), robotics (He et al., 2019), linguistics (Chu and Spinney, 2018), and video coding (Kamble et al., 2018). In addition, FSM theory plays an im-portant role in many areas of modern computer sci-ence (Yan et al., 2019a).

State space optimization of FSMs is an important issue in FSM theory, and is of great significance in practical applications in engineering because the memory space needed for hardware realization grows exponentially with the increase of the number of states of FSMs. The problem of state space optimiza-tion of FSMs refers to the minimization of the number of the states of an FSM with the resulting FSM re-taining the same functions as the original FSM.

The optimization of FSMs has been extensively studied from various aspects, including different methods used for the same type of FSM, similar ap-proaches used for different types of FSM, and some with different emphases on some particular issues. For lattice-valued multiset FSMs, Li and Pedrycz (2007) and Wang and Li (2018) established

Frontiers of Information Technology & Electronic Engineering

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ISSN 2095-9184 (print); ISSN 2095-9230 (online)

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‡ Corresponding author * Project supported by the National Natural Science Foundation of China (Nos. U1804150, 62073124, and 61973175)

ORCID: Jumei YUE, https://orcid.org/0000-0003-1474-0088; Yongyi YAN, https://orcid.org/0000-0002-7181-5894 © Zhejiang University Press 2021


equivalences of nondeterministic lattice multiset FSMs and deterministic ones. Based on the equiva-lences, they further proposed effective algorithms to produce a minimal substitution of a given lattice multiset FSM. For Mealy and Moore FSMs, Solov’ev (2011, 2017) used two methods to study the state space optimization of Mealy FSMs by merging in-ternal states and using output variables of state as-signment based on structural models of FSMs. It was shown that the methods can reduce the area of FSM implementation for all families of field programmable gate arrays (FPGAs) of various manufacturers. The minimization of Moore FSMs was investigated by gluing states and representing an FSM in the form of a transition list (Solov’ev, 2010). With regard to the state space optimization of incompletely specified FSMs, the way of merging internal states used origi-nally for Mealy FSM minimization is applied to this type of FSM by Klimowicz and Solov’ev (2013). The proposed minimization methods can considerably reduce the number of internal states and the number of FSM transitions. There are other studies on state space optimization with a specific emphasis, such as state reduction (Gören and Ferguson, 2007), exact minimization (Gören and Ferguson, 2002), and di-mensionality minimization (Perkowski et al., 2001). On the state space optimization of deterministic FSMs (DFSMs), there are some excellent methods, for example, using backward depth information (Liu et al., 2016), inspired by Brzozowski’s algorithm (García et al., 2014), the double reversal minimization algorithm (de Parga et al., 2013), and the complex method (Solov’ev, 2014). Also, there are some tech-niques with different focuses, for example, edge minimization (Melnikov, 2010), power consumption (Grzes and Solov’ev, 2015), and transition minimiza-tion (Dahmoune et al., 2014).

Most studies, like those mentioned above, are computer-based in nature. This can be either incon-venient or even unable to reveal the mathematical meanings of the optimization of FSMs. This moti-vates further consideration of the state space optimi-zation problem from the standpoint of control science, where FSMs are viewed as logical dynamic systems, aiming to reveal the mathematical aspects of state space optimization. The research is based on the state transition dynamic equation of FSMs established under the framework in control theory in recent years

(Xu and Hong, 2012; Yan et al., 2015). In Xu and Hong (2012) and Yan et al. (2015), the dynamics of an FSM is formulated as Eqs. (1) and (2), respectively:

( 1) ( ) ( ),

( ) ( ) ( ),

t t t

t t t

x Fu x

y Hu x (1)

and

0

0

( 1) ( ),

( ) ( ),

t

t

t t

t t

x T x u

y K x u (2)

where x(t+1) is the state at time step t+1, x0 is an initial state, u(t) is an input string of length t, y(t) is the output at time step t, and Tt is a matrix called a dynamical matrix of an FSM. Eqs. (1) and (2) share similarities in results while having differences con-ceptually. In Eq. (1) the state of an FSM is defined as a vector x(t)=(x1(t), x2(t), …, xn(t))

T, where xi(t) (i=1, 2, …, n) is the number of different paths by which the FSM can reach state xi with an input string of length t−1, while in Eq. (2) state xi is simply defined as a

vector in ( i

n is the column vector corresponding to

the ith column of unit matrix In). Eq. (1) is suitable for both DFSMs and nondeterministic FSMs (NFSMs), and describes the 1-step dynamics, while Eq. (2) is suitable only for DFSMs, and formulates the t-step dynamics. Thus, Eq. (1) excels in modeling FSMs, and Eq. (2) has advantages in expressing the dynam-ics. Based on Eq. (2), one can analyze and synthesize FSMs like studying dynamic systems using control theory. The ideas such as feedback (Han and Chen, 2018; Zhang ZP et al., 2018b), the concepts such as controllability, observability, reachability, and stabi-lization (Yan et al., 2016; Zhang KZ and Zhang, 2016; Han et al., 2018; Zhang ZP et al., 2018a), and the methods such as state feedback control, corrective control, and fault tolerant control (Gao et al., 2017; Lu et al., 2017; Zhang ZP et al., 2017) have been trans-planted to FSMs.

The dynamic model of FSMs (2) is adopted in this study to investigate the state space optimization of FSMs because those considered here are limited to DFSMs since DFSMs are simple and easy to imple-ment in engineering, and NFSMs are equivalent to DFSMs (Gören and Ferguson, 2007; Li and Pedrycz, 2007). Here “equivalent” means that for any NFSM


there exists a DFSM with the same function as the nondeterministic one.

The main contributions of this paper are as fol-lows. Taking the concept of equilibrium points in control theory, the concepts of t-equivalent states and t-source equivalent states of FSMs are proposed. Based on the state transition dynamic equation (2), several necessary and sufficient conditions are pro-posed for t-equivalent states and t-source equivalent states. These can be analogized to the necessary and sufficient conditions of equilibrium points of dynamic systems in control theory and thus explain mathe-matically the optimization problem. Two methods are designed to find all the t-equivalent states and t-source equivalent states of FSMs. These two methods are further applied to develop two methods of reducing FSMs. Unlike the computer-based algo-rithms proposed by researchers in computer science, the proposed methods are independent of computers and can be implemented mathematically with pen and paper; of course, they can also be easily conducted on computers. 2 Preliminaries and problem description

In this section we first briefly review the re-search tool, the semi-tensor product (STP) of matrices, and the research object, FSMs, and then give a prob-lem description.

2.1 Semi-tensor product of matrices

Definition 1 (STP of matrices) (Cheng et al., 2012) For any two matrices Mm×n and Np×q, their STP, de-noted by ,M N is defined as

/ /: ( )( ),s n s p M N M I N I

where s is the least common multiple of n and p, and is the Kronecker product of matrices. Remark 1 In practice, a special form of the STP where n and p are of multiple relations is often used, which is given as follows:

1. Let X be a row vector of dimension np, and Y be a column vector with dimension p. Split X into p blocks X1, X2, …, Xp of equal size, which are 1×n row vectors. The STP, denoted by , is defined as

1

T T T

1

= ,

( ) ,

pn

i ii

pn

i ii

y

y

X Y X

Y X X

where iy is the ith component of Y.

2. Let A and B be m×n and p×q matrices, re-spectively. If either n is a factor of p or p is a factor of n, then the STP of A and B, denoted by

{ } ,ij C C A B is defined as follows: C consists

of m×q blocks and each block is defined as

= , 1,2, , , 1,2, , ,i

ij j i m j q C A B

where Ai is the ith row of A and Bj is the jth column of B. Remark 2 (1) STP generalizes the conventional multiplication of matrices. It reduces to the latter when n=p. In this study the multiplication of matrices defaults to the STP of matrices, and the symbol “” is omitted except for special emphases. (2) The STP of matrices not only retains almost all the major properties of the conventional matrix multiplication (for instance, the associative law is that for Am×n, Bp×q, and Cr×s, we have ( ) ( )),A B C = A B C but

also overcomes some inherent defects of the latter. Let us look at an interesting phenomenon of the conventional matrix multiplication: “legal opera-

tions” leads to “illegal outcomes.” Let X, Y, Z, Wn

be column vectors. Since YTZ is a real number, we have that (XYT)(ZWT)=X(YTZ)WT=(YTZ)(XWT) is an m×n matrix. Continuing application of the associa-tivity leads to (YTZ)(XWT)=YT(ZX)WT. Now there is a question “what is the term ZX?” Clearly, it is an illegal term. This shows that the conventional multi-plication of matrices is not completely “compatible.” However, under the framework of the STP of matrices, such an abnormal phenomenon turns normal. (3) The STP of matrices has been developed well and applied successfully in many areas, such as Boolean networks (Yan et al., 2019c; Yue et al., 2019a; Zhang QL et al., 2020), applied graph theory (Yan et al., 2019b), finite state machines (Yue et al., 2019b, 2020a, 2020b), and matrix theory (Yue and Yan, 2019).


2.2 Finite state machines

Definition 2 (FSMs) (Yan et al., 2019a) An FSM is a five-tuple M=(X, E, f, x0, XF), where X and E denote two finite sets of states and input symbols, respec-tively, f is a state transition function from X×E to X, x0 is an initial state at which M begins to run, and XF is a set of final states indicating that M accepts an input string if M reads the input string and stops at one state of XF. Fig. 1a is an example, where X={x1, x2, x3, x4, x5, x6}, E={0, 1}, x0=x1, XF={x6}.

FSMs are of two kinds, deterministic and non-deterministic. An FSM M is said to be deterministic if |f(x, e)|≤1 holds for each xX and eE, where |f(x, e)| denotes the number of states to which M is moved by input e from state x; otherwise, M is called an NFSM. Since NFSMs are equivalent to DFSMs, and DFSMs are used widely in practice, the FSMs considered in this study are limited to DFSMs.

The dynamical evolution that FSM M reads an input string e=e1e2…etE* at state xX is defined as

1 2( , ) ( ( ( ( , ), ) ), ),tf e x f f f f x e e e (3)

where E* denotes the set of finite strings on E={e1, e2, …, em}. The meaning of Eq. (3) is that M starts in state x and makes a transition to state f(x, e1)X if event e1 occurs at state x. Next, if an event e2 occurs at

the new state f(x, e1), M transfers to state f(f(x, e1), e2), and so on.

2.3 Problem description

An FSM is originally a concept in the field of computer science, where the state evolution is ex-pressed by diagrams, tables, or discrete functions. These kinds of representations are simple and intui-tive, and provide the basis for designing algorithms to accomplish various kinds of particular functionalities. However, they are not easy for studying the state evolution of FSMs mathematically, i.e., not like in-vestigating the dynamics of linear and nonlinear sys-tems by manipulating their dynamic equations.

In recent years the state transition dynamics of an FSM is formulated as a dynamic equation (4) (Yan et al., 2015), called the state transition dynamic equation, with which one can analyze the dynamics of FSMs like using dynamic equations to investigate the behaviors of dynamic systems in the area of control theory:

0( 1) ( ). tt tx T x u (4)

In Eq. (4) both the states x(t) and x0 and input

string u(t) are vectors. Specifically, for FSM M=(X, E, f, x0, XF), where X={x1, x2, …, xn} and E={e1, e2, …,

em}, state xi is identified with in , and input symbol ej

with jm ; i

n and jm are called the vector forms of

state xi and input symbol ej, respectively, where qp is

the column vector corresponding to the qth column of the p×p unit matrix Ip. For an input string

u(t)=e1e2…et, its vector form is 1t jj m . This repre-

sentation is referred to as the vectorization of M. This is intended to introduce the STP of matrices to model the dynamics of M as a dynamic equation. In this study the vector forms of state xi and input symbol ej are used interchangeably with xi and ej for the con-venience of use.

The aim of this paper is to find, based on the dynamic equation (4), the simplest substitutions of FSMs by following the style of control theory. A simplest substitution of an FSM is an FSM that achieves the same function with the minimum num-ber of states, for example, generation of the same language. For this purpose, we propose the concepts

x1

0

1

1

0

0

1

11

0

0

0, 1

(a)

x2

x3

x4

x5

x6

x1

0

1

1

0

1

0, 1

0

00, 1

(b)

x2

x3

x4

x6

Fig. 1 The original finite state machine (a) and merged finite state machine (b)


of t-string set between states, t-equivalent states and t-source equivalent states; we explore some mathe-matics of t-equivalent states, and further design sev-eral algebraic methods to find the simplest substitu-tions of FSMs. In addition, an open question is posed, by which the proposed optimization method can be further optimized to an unattended one.

The following are the notations used in this

paper: in , the column vector corresponding to the ith

column of the unit matrix In; Coli(A), the ith column of matrix A; |w|, the length of string w, i.e., the number of symbols contained in w; α_β, the connection of strings α and β; S1−S2, the subtraction of sets S1 and S2, i.e., S1−S2={x|xS1, xS2}; , the empty set. 3 Main results

In this section we present the main results of this paper in two parts. The first part includes the theo-retical results containing the concepts and mathe-matical formulations of t-equivalent states and t-source equivalent states, which are the basis of de-signing the methods of reducing FSMs presented in the second part.

3.1 Theories

Definition 3 (t-string set between states) Let M=(X, E, f, x0, XF) be an FSM, where X={x1, x2, …, xn} and E={e1, e2, …, em}. The set of input strings of length t moving M to xj from xi is called the t-string set from xi to xj, denoted by St(xi→xj), i.e.,

*( )={ | , | |= , ( , ) }. t i j i jS x x E t f x x

Definition 4 (t-equivalent states and t-different states)

States 1i

x and 2i

x are called t-equivalent states if the

following Eq. (5) holds for any 1 1

( ) t i jS x x

2 2( ) :t i jS x x

1 2( , ) ( , ).i if x f x (5)

Otherwise, they are called t-different states; that is,

there is an 1 1 2 2

( ) ( ) t i j t i jS x x S x x such that

1 2( , ) ( , ).i if x f x

Remark 3 (1) Since 1 1

( , )i jf x x and 2

( , ) if x

2,jx the t-equivalence of states

1ix and

2ix implies

that 1 2j jx x in Definition 4. (2) Clearly, as a special

case, state xi is t-equivalent to itself. (3) The t-equivalence of states satisfies the following three properties: reflexivity, symmetry, and transitivity; therefore, it is an equivalence relation on the set of states X. (4) If states

1 2, , ,

si i ix x x are t-equivalent,

they have exactly the same functions. In other words, they exhibit exactly the same behaviors in the dy-namic evolutions of FSMs. (5) To reduce the state space of FSMs, it is sufficient to find the equivalent states and merge them to one.

For the convenience of description in the sequel, we introduce a partition of matrices.

r-column equipartition of matrices: let M be a matrix of size p×qr. The r-column equipartition of M refers to the following partition that divides M into r equally sized blocks:

1[Blk ( ), ,Blk ( ), ,Blk ( )],i r M M M M

where

( 1) 1 ( 1) 2Blk ( ) [Col ( ),Col ( ),

,Col ( )], 1,2,..., .

i i q i q

iq i rM M M

M

By the definition of the STP of matrices and di-

rect calculation, it is easy to obtain the following proposition, which will be used extensively in the proofs of the theorems of this paper: Proposition 1 Let M be a matrix of size p×qr. Then

Blk ( ),ir iM M (6)

where Blki(M) (i=1, 2, …, r) is the ith block of r-column equipartition of M. Theorem 1 (Equivalent condition of states) Let Eq. (7) be the state transition dynamic equation of FSM M=(X, E, f, x0, XF), where X={x1, x2, …, xn} and E={e1, e2, …, em}:

( 1) ( )tit t x T x u . (7)

Then 1i

x and 2i

x are t-equivalent if and only if

1 2Blk ( ) Blk ( ).t t

i iT T


Proof (Necessity) Since states 1i

x and 2i

x are

t-equivalent, it follows from Definition 4 that the


( ) t i jS x x

2 2( ) :t i jS x x

1 2( , ) ( , ).i if x f x (8)

By Eq. (7), Eq. (8) can be rewritten as

1 2,t t

i ix x T T (9)

where ω is the input string u(t) of Eq. (7).

Note that the vector forms of 1i

x and 2i

x are 1in

and 2 ,in respectively. Substituting the vector forms

into Eq. (9) yields

1 2 .i it tn n T T (10)

Combining Eq. (6) with Eq. (10), we have that

1 2Blk ( ) Blk ( ) .t t

i i T T The arbitrariness of ω

guarantees that 1 2

Blk ( ) Blk ( ).t ti iT T The necessity

is proved. (Sufficiency) The known condition is

1 2Blk ( ) Blk ( ).t t

i iT T We prove the sufficiency by

contradiction. Assume 1i

x and 2i

x are not t-equivalent.

By Definition 4, there exists an 1 1

( ) t i jS x x

2 2( )t i jS x x such that

1 2( , ) ( , ).i if x f x (11)

Using Eq. (7), we know that the left side of in-

equality (11) is

1,t

ix T (12)

and the right side of inequality (11) is

2.t

ix T (13)

Substituting the vector forms of 1i

x and 2i

x , 1in

and 2in , into expressions (12) and (13), respectively,

we have 1 ,it

nT (14)

and 2 .it

n T (15)

According to Proposition 1, expressions (14) and

(15) are 1

Blk ( )ti T and

2Blk ( ) ,t

i T respectively.

Thus, inequality (11) can be rephrased as

1 2Blk ( ) Blk ( ) .t t

i i T T

On the other hand, let the vector form of be

1 . t

t i li m m Then Proposition 1 shows that the lth

block of 1

Blk ( )ti T is different from the lth block of

2Blk ( )t

i T , i.e.,

1 2Blk ( ) Blk ( ).t t

i iT T

This contradicts the known condition

1 2Blk ( ) Blk ( ).t t

i iT T The proof of sufficiency is

completed. Corollary 1 For the FSM described in Theorem 1, if states

1ix and

2ix are t-equivalent, they are

(t+k)-equivalent (k≥1).

Proof Since states 1i

x and 2i

x are t-equivalent, it

follows from Theorem 1 and Definition 4 that the


( ) t i jS x x

2 2( ) :t i jS x x

1 2Blk ( ) Blk ( ).t t

i iT T (16)

By Proposition 1, we know that Eq. (16) is

equivalent to

1 2.t t

i ix xT T (17)

For any input symbol eE, we consider the

string 1 1 2 21 1_ ( ) ( ). t i j t i je S x x S x x

Multiplying both sides of Eq. (17) by T on the left and by ω_e on the right (for convenience, in the following we use ω_e to denote the vector form of the string ω_e), we obtain


1 2_ _ ,t t

i ix e x e TT TT (18)

i.e.,

1 2

+1 +1_ _ .t ti ix e x e T T (19)

According to Eq. (7), Eq. (19) can be rewritten as

1 2( , _ ) ( , _ ).i if x e f x e

Since ω_e is arbitrary, it follows from Definition

4 that 1i

x and 2i

x are (t+1)-equivalent. Setting t=t+1,

we have that 1i

x and 2i

x are (t+2)-equivalent. Fol-

lowing this procedure, the proof can be completed. Definition 5 (t-source equivalent states) Let

1ix and

2ix be two states of an FSM.

1ix and

2ix are t-source

equivalent states if 1i

x and 2i

x are t-equivalent, but

not (t−1)-equivalent (t≥2). If 1i

x and 2i

x are

1-equivalent, they are called 1-source equivalent states. Theorem 2 (Necessary condition of (t+1)-equivalent states) Let M=(X, E, f, x0, XF) be an FSM, where X={x1, x2, …, xn} and E={e1, e2, …, em}. If there are two (t+1)-equivalent states

1ix and

2ix , there exist at

least two t-equivalent states.

Proof Since 1i

x and 2i

x are (t+1)-equivalent, by

Definition 4, the following Eq. (20) holds for any

1 1 2 2+1 +1_ ( ) ( ), t i j t i je S x x S x x where |ω|=t:

1 2( , _ ) ( , _ )i if x e f x e . (20)

The expansion of Eq. (20) is

1 2( ( , ), ) ( ( , ), ).i if f x e f f x e (21)

Suppose that

1 1( , ) ,i kf x e x (22)

2 2( , ) .i kf x e x (23)

Setting Eqs. (22) and (23) into Eq. (21), we obtain

1 2( , ) ( , ).k kf x f x

Because ω (|ω|=t) is arbitrary, it follows from Defi-nition 4 that

1kx and 2kx are t-equivalent. This proves

the theorem.

The following corollaries are useful conse-quences of Theorem 2 and will be used in the state optimization of FSMs presented in the following subsection: Corollary 2 Let M=(X, E, f, x0, XF) be an FSM, where X={x1, x2, …, xn}. If M contains no t-equivalent states, there are no (t+1)-equivalent states.

Proof (Deduction to absurdity) Assume that 1i

x and

2ix are (t+1)-equivalent states. According to Theorem

2, we know that there exist at least two t-equivalent states. This contradicts the known condition that M contains no t-equivalent states. Thus, Corollary 2 holds.

Based on Corollary 2 we have the following result: Corollary 3 If an FSM contains no 1-equivalent states, then it is the simplest substitution of itself. Remark 4 Theorem 1 and Corollaries 1–3 show that the dynamical matrix Tt (Eq. (7)) completely depicts the properties of the states of FSM: states are either t-equivalent or t-different. Corollaries 2 and 3 provide a theoretical basis for the termination condition of designing methods to reduce FSMs.

3.2 Methods

In this subsection we present several algebraic methods of finding t-equivalent states and t-source equivalent states and of reducing FSMs.

The following Method 1 of finding all the t -equivalent states of FSMs is designed based on the sufficiency and necessity of Theorem 1: Method 1 (Finding t-equivalent states) Let x(t+1)=Ttxiu(t) be the state transition dynamic equa-tion of FSM M=(X, E, f, x0, XF), where X={x1, x2, …, xn}. Taking the following procedure, we can obtain all the t-equivalent states of M:

Step 1: divide matrix Tt using the n-column eq-uipartition, and denote the ith block by Blki(T

t), i=1, 2, …, n.

Step 2: check if there are identical blocks in {Blki(T

t)|i=1, 2, …, n}. If not, M contains no t-equivalent states and the method terminates. Oth-erwise, construct the set K={(i1, i2, …, is)|2≤s≤n, Blki(T

t) are identical, j=1, 2, …, s}. Step 3: the set of t-equivalent states of M is

1 2 1 2( , , , ) | ( , , , ) .

st i i i sX x x x i i i K


The following Method 2 is to find the t-source equivalent states of FSMs: Method 2 (Finding t-source equivalent states) Let x(t+1)=Ttxiu(t) be the state transition dynamic equa-tion of FSM M=(X, E, f, x0, XF), where X={x1, x2, …, xn}. The following steps yield the set of t-source equivalent states of M:

Step 1: use Method 1 to obtain the set of t-equivalent states of M, Xt.

Step 2: use Method 1 to obtain the set of (t−1)-equivalent states of M, Xt−1.

Step 3: obtain the set of t-source equivalent states of M, Xot=Xt−Xt−1.

Next we consider how to reduce the state space of FSM; we start with a preparation of regulating an operation of merging states into one state.

Merging of states of FSMs: let X be the state set of an FSM M, where X={x1, x2, …, xn}. Let

1 2{ , , , }

si i iX x x x be a subset of X. The merging of

states refers to the process of merging states in X′ into one and moving the related “edges” accordingly in the state transition diagram of M. We use an example, as shown in Fig. 1, to illustrate it, where X={x1, x2, x3, x4,

x5, x6}, 2 5={ , }.X x x

Method 3 (State minimization of FSMs) Let M be an FSM, where the state set is X={x1, x2, …, xn}. Take the following steps and we can obtain the simplest substitution of M:

Step 1: set t=1 and Box=. Step 2: use Method 2 to obtain the set of t-source

equivalent states of M, denoted by Xot. If there are no t-source equivalent states, let Xot=.

Step 3: set Box=BoxXot. Step 4: check whether Xot is empty or not. If yes,

go to step 5; otherwise, set t=t+1, and go to step 2. Step 5: merge the states in each boxBox. The

resulting FSM is the simplest substitution of M. Remark 5 The dimension of Tt is n×mtn, which shows that the size of Tt increases exponentially with m and t. Thus, Tt is very large when m or t is large. This causes a heavy burden on the memory of com-puters during the use of Method 3 to reduce FSMs (Note that Tt appears in Method 1, Method 1 is used in Method 2, and Method 2 is used in Method 3). To avoid this situation, Method 3 can be further improved. Method 4 (Improvement of Method 3)

Step 1: use Method 1 to obtain the set of 1-equivalent states, X1. If X1 does not exist, M is a simplest FSM and the method terminates; otherwise, go to step 2.

Step 2: merge the states in X1. Denote the re-sulting FSM by Mnew.

Step 3: set M=Mnew and go to step 1. Remark 6 In Method 4, the iterative merging of the 1-equivalent states is used to avoid the appearance of Tt and prevent the heavy burden on computer memory.

3.3 Complexity analysis

In this subsection we analyze the computational complexity of the presented minimization method (Method 4) from the two aspects of time complexity and space complexity. Method 4 is an improved ver-sion of Method 3; Method 3 is conducted based on Methods 1 and 2. Therefore, we need only to discuss the complexities of Methods 1–3. Consider an FSM M=(X, E, f, x0, XF), where X={x1, x2, …, xn} and E={e1, e2, …, em}.

Time complexity: Methods 1 and 2 have no circulation iteration, and thus the time complexity is O(1). In Method 3, if M itself is the simplest FSM, the method comes to an end without iteration. Suppose M is not the simplest one. The number of t-source equivalent states is n−1 in the worst situation, and for this reason the case of Xot= occurs, thus causing Method 3 to iterate n−1 times. Therefore, the method iterates n−1 times at most. The time complexity is O(n).

For Method 4, since the number of states of an FMS is at least one, the number of iterations of the method is no larger than n−1. Therefore, the time complexity is of polynomial time O(n).

Space complexity: in Method 1, matrix Tt of size n×mtn and matrices Blki(T

t), i=1, 2, …, n, need to be stored in step 1. The sum of the sizes of these n ma-trices is n×mtn. Thus, step 1 requires a total of 2n2mt storage units. The set K needs to be stored in step 2; the number of elements of K is the combinatorial

number 2C ( 1) n n n in the worst case. The set Xt in

step 3 has also n(n−1) elements at most. Therefore, the space complexity of Method 1 is O(n2mt). Method 2 uses Method 1 to find the t-source equivalent states of M, and no additional storage is required. The space complexity of Method 2 is also O(n2mt). The storage


needed by Method 3 lies in step 2, in which Method 2 is applied to obtain the t-source equivalent states of M. In the worst situation, step 2 is iterated n times, and thus the space complexity is nO(n2mt)=O(n3mt).

For Method 4, only step 1 needs memory. Step 1 is to find the 1-source equivalent states of M using Method 1. Therefore, the space complexity of Method 3 can be obtained by setting t=1 in the space com-plexity of Method 1, O(n2mt), and the result is of polynomial time O(n2m). 4 Discussions (an open question)

Existing results of FSM minimization can roughly be classified into four categories according to research methods (Martinek, 2018): merging of states, refinement of the state set, trim of FSMs, and dy-namic minimization. Most of them are computer- based algorithms, which focus on the minimization method and stress the computation gain. They are efficient for engineering problems; however, they fail to describe the minimization in a mathematical manner.

Compared with most previous work, although our work is as simple and efficient as them, more importantly, they differ in concept, purpose, approach, and the focus on both the theoretical and algorithmic results. The existing work frequently develops new algorithms to gain computational improvement; of course, most are supported by theoretical results. However, they are not good at explaining the math-ematical meaning of FSM minimization. We recon-sider the problem from the standpoint of control the-ory, where FSMs are treated as logical dynamic sys-tems, aiming to reveal the mathematical meaning of the minimization.

The advantage of this study is the mathematical formulation and mathematical methods for simplify-ing FSMs. These explain the mathematics of the simplification problem.

Method 4 can be further improved. Let Tnew and T be the dynamical matrices of the new FSM Mnew and the original FSM M in Method 4, respectively. Denote by Tnew=g(T) the relation between Tnew and T. If Tnew=g(T) is known, then Tnew can be obtained directly from T without replacing Mnew by M in Method 4. Consequently, Method 4 can be optimized

to run offline. Method 5 (Unattended version of Method 4)

Step 1: split matrix T using n-column equiparti-tion and denote by Blki(T) the ith block, i=1, 2, …, n.

Step 2: check whether there are identical blocks in {Blki(T)|i=1, 2, …, n}. If not, M is a simplest FSM and stops. Otherwise, set

1 21 {( , , , ) |si i iX x x x

.are id2 entil ca ,, ,lB k ( ) 1,2, }jis n j s T

Step 3: merge the states in X1 and denote by Mnew the resulting FSM.

Step 4: set T=Tnew=g(T), and go to step 1. An open question: does the relation Tnew=g(T)

between Tnew and T exist? If yes, what is it? This problem is under our study. 5 Correctness and effectiveness analysis

In this section we use an example in Chen (2007) to verify the correctness and effectiveness of the methods presented in this paper. Consider the FSM M=(X, E, f, x0, XF), as shown in Fig. 2, which recog-nizes the language over alphabet {0, 1} that contains no “00” as its substring, where X={x0, x1, x2, x3, x4, x5}, E={0, 1}, x0=x1, and XF={x1, x2, x4, x5}.

Note that the main results of this paper have the relations shown in Fig. 3, which suggests that veri-fying the effectiveness and correctness of Method 4 will suffice.

Step 1: run step 1 of Method 4. Use Method 1 to obtain the set of 1-equivalent states X1. The state transition dynamic equation is x(t+1)=Ttx1u(t), where the state set X={x1, x2, …, x5} and T=δ5[2, 5, 3, 5, 3, 3, 3, 5, 4, 5].

Step 1.1: run step 1 of Method 1. Divide T into 5-column equipartition:

1 5 2 5

3 5 4 5

5 5

Blk ( ) [2,5], Blk ( ) [3,5],

Blk ( ) [3,3], Blk ( ) [3,5],

Blk ( ) [4,5].

T T

T T

T

Step 1.2: run step 2 of Method 1. In {Blki(T)|i=1, 2, …, 5}, Blk2(T)=Blk4(T). We have K={(2, 4)}.

Step 1.3: run step 3 of Method 1. The set of 1-equivalent states of M is X1={(x2, x4)}.

Step 2: run step 2 of Method 4. Merge the states x2 and x4. The resulting FSM Mnew is shown in Fig. 4.


Step 3: run step 3 of Method 4. Set M=Mnew and go to step 1 of Method 4.

Step 4: run step 1 of Method 4. Use Method 1 to obtain the set of 1-equivalent states of M, X1. Now the state transition dynamic equation is x(t+1)=Ttx1u(t), where X={x1, x2, x3, x4} and T=δ4[2, 4, 3, 4, 3, 3, 2, 4].


1 4 2 4

3 4 4 4

Blk ( ) [2,4], Blk ( ) [3,4],

Blk ( ) [3,3], Blk ( ) [2,4].

T T

T T

Step 4.2: run step 2 of Method 1. In {Blki(T)|i=1,

2, 3, 4}, Blk1(T)=Blk4(T). We have K={(1, 4)}. Step 4.3: run step 3 of Method 1. The set of

1-equivalent states of M is X1={(x1, x4)}.

Step 5: run step 2 of Method 4. Merge the states x1 and x4. The resulting FSM Mnew is as shown in Fig. 5.

Step 6: run step 3 of Method 4. Set M=Mnew and go to step 1 of Method 4.

Step 7: run step 1 of Method 4. Use Method 1 to obtain the set of 1-equivalent states of M, X1. Now the state transition dynamic equation of M is x(t+1)= Ttx1u(t), where X={x1, x2, x3} and T=δ3[2, 1, 3, 1, 3, 3].


1 3 2 3

3 3

Blk ( ) [2,1], Blk ( ) [3,1],

Blk ( ) [3,3].

T T

T

Step 7.2: run step 2 of Method 1. There are no

identical blocks in {Blki(T)|i=1, 2, 3}. Then M has no 1-equivalent states. Method 1 is terminated. X1 does not exist, and M is a simplest FSM. Method 4 stops.

It is easy to verify that the reduced FSM (Fig. 5) has the same functions as the original FSM (Fig. 2). Both recognize the language on alphabet {0, 1} in which there is no substring “00.” This shows the correctness and effectiveness of the results of this paper. 6 Conclusions

FSMs can be viewed as logical systems. In this paper, we use the concepts, ideas, and methods of

0 x3

1

0, 1

0

0

1

01

x4

x2

x1

x5

Fig. 2 Finite state machine to be reduced

Method 4

Method 2

Corollary 3

Theories

Methods

Proposition 1

Theorem 1

Corollary 1

Corollary 2

Theorem 2

Method 1

Method 3

Fig. 3 Logical relations among the main results of this paper

0x3

0,1

0

1

1

x2x1

Fig. 5 Finite state machine produced by merging 2-equivalent states x1 and x4

0 x3

1

0, 1

0

01

1

x2

x1

x5

Fig. 4 Finite state machine produced by merging 1-equivalent states x2 and x4


system science to reconsider the problem of state space optimization of FSMs. Two general kinds of equivalent states of FSMs, t-equivalent states and t-source equivalent states, are proposed by following the notion of the equilibrium points of dynamic sys-tems in control theory. Further we explore some mathematics of these two kinds of equivalent states, including properties and a necessary and sufficient condition of t-equivalent states, and the relation be-tween the t-equivalent and (t+1)-equivalent states. Based on these results, several algebraic methods are developed to find all the t-equivalent states and to reduce FSMs. The idea and methods proposed in this paper may provide a new angle and means to analyze and synthesize FSMs mathematically.

Contributors Jumei YUE raised the research questions and the ideas to

solve them. Yongyi YAN and Zengqiang CHEN guided Jumei YUE and He DENG to carry out the research in the whole process. Jumei YUE drafted the manuscript. He DENG helped organize the manuscript. Yongyi YAN and Zengqiang CHEN revised and finalized the paper.

Acknowledgements

The authors would like to express their sincere thanks for the advice given by Assistant Prof. Xiangru XU and Prof. Yiguang HONG.

Compliance with ethics guidelines

Jumei YUE, Yongyi YAN, Zengqiang CHEN, and He DENG declare that they have no conflict of interest.

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